As modules are added to an overall model, there is an increasing hazard for misspecification. Methods that help us carry out Bayesian analysis on misspecified models have been in development for some time without explicit consideration of modularisation.
We divide these into three classes. Methods which temper the likelihood lead to power posteriors, [1]}, [2]}, [3]}, methods which use bootstrapping in a Bayesian setting, including Weighted likelihood Bootstrap [4]}, [5]}, [6]}, the Posterior Bootstrap [7]} and BayesBag [8]}, [9]}, and methods which replace the likelihood with some more general loss function mediating data and parameter, including PAC-Bayes [10]}, [11]}, [12]}, [13]}, Gibbs posteriors [11]}, [15]} and Generalized Bayes [16]}, [17]}
are relevant in the multi-modular setting.
